A Bayesian network is one type of a graphical statistical model that encodes probabilistic relationships among variables of interest. Over the last decade, the Bayesian network has become a popular representation for encoding uncertain expert knowledge in expert systems. When used in conjunction with statistical techniques, the graphical model has several advantages for data analysis. Because the model encodes dependencies among all variables, it readily handles situations where some data entries are missing. A graphical model, such as a Bayesian network, can be used to learn causal relationships, and hence can be used to gain understanding about a problem domain and to predict the consequences of intervention. Because the model has both a causal and probabilistic semantics, it is an ideal representation for combining prior knowledge (which often comes in causal form) and data. Additionally, Bayesian statistical methods in conjunction with Bayesian networks offer an efficient and principled approach for avoiding the over fitting of data.
Graphical statistical models facilitate probability theory through the utilization of graph theory. This allows for a method of dealing with uncertainty while reducing complexity. The modularity of a graphical model permits representation of complex systems by utilizing less complex elements. The connections and relationships of individual elements are identified by the probability theory, while the elements themselves are constructed by the graph theory. Utilizing graphics also provides a much more intuitive human interface to difficult problems.
Nodes of a probabilistic graphical model represent random variables. Their connectivity can indicate associative qualities such as dependence and independence and the like. If no connectivity (i.e., “arcs”) is present, this represents conditional independence assumptions, providing a representation of joint probability distributions. Graphical models can be “directed” or “undirected” depending on how they are constructed. Undirected graphical models have a more simplistic definition of independence, while directed graphical models are more complex by nature. Bayesian or “Belief” networks (BN) are included in the directed category and are utilized extensively in statistics and artificial intelligence to show causality between elements or “nodes.” They are also highly beneficial in supplying “inferences.” That is, they are able to infer information based on a posterior probability (i.e., “likelihood”) utilizing Bayes' rule. Therefore, for a given outcome, its cause can be probabilistically deduced utilizing a directed graphical model.
An often used type of Bayesian network is known as a conditional Gaussian (CG) network and is a graphical model that encodes a conditional Gaussian distribution for variables of a domain. A CG model is a directed graphical model with both discrete and continuous variables. For a directed graphical model, the structural relations between variables X=(X1, . . . , Xm) are represented by a directed acyclic graph (DAG), where each node represents a variable Xv, and directed edges represent direct influence from variables represented by parent variables Xpa(v). A CG model is characterized by: (i) the graphical DAG structure has no discrete variable with a continuous parent variable, (ii) the joint model can be represented by local conditional models p(Xv|Xpa(v), (iii) the local models for discrete variables (given discrete parents) are defined by conditional multinomial distributions—one for each configuration of possible values for the discrete parents, and (iv) the local models for continuous variables (given continuous and discrete parents) are defined by conditional Gaussian regressions—one for each configuration of possible values for discrete parents. Because of the usefulness of this type of model, being able to efficiently facilitate in determining parameters for constructing a conditional Gaussian graphical model is highly desirable.